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Viggo Brun Prize Winner 2018

For fundamental contributions to the theory of higher categories, with applications to quantum field theory, representation theory, algebraic geometry, and geometric topology, and for the development of higher Morita theory and enriched ∞-categories.

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Rune Haugseng (born 1984) is a Norwegian mathematician with BA and MA degrees from Cambridge University. He earned a PhD from Massachusetts Institute of Technology in 2013 with Haynes Miller as supervisor. He has held post-doc positions at the Max Planck Institute for Mathematics in Bonn and at the University of Copenhagen. Starting in the fall of 2019 he will be an associate professor at NTNU.

Haugseng studies higher categories. These are structures that occur in many places in mathematics where some property typically only holds up to (a hierarchy of) equivalences that have to fit together.

As an example, parts of Haugseng’s work give contributions to topological quantum field theories. Quantum field theories associate to space-time (geometry) its state space (algebra) with a time-evolution operator. However, the global structure should be governed by the local information. To make this into a mathematically acceptable theory has proven to be a challenge, keeping mathematicians busy for decades. The problem is that many things interact in a way that yields an enormously complicated system of coherence problems. Parts of Haugseng’s work also have direct connections to derived algebraic geometry where similar phenomena occur.

Among Haugseng’s many important works, there are two monumental contributions: A groundbreaking 140-page joint paper with David Gepner, published in Advances in Mathematics, develops the theory of enriched ∞-categories, and in a 100-page paper in Geometry & Topology, Haugseng solves the central problem of how to construct a higher version of the Morita category of associative algebras, bimodules, and bimodule homomorphisms. Haugseng’s set-up provides a context for factorization homology, a topological variant of Beilinson–Drinfeld’s chiral homology proposed by Jacob Lurie. Given an En algebra (an algebra where commutativity holds up to “level n”) factorization homology gives rise to a topological quantum field theory for n-dimensional manifolds.

Haugseng’s pioneering work provides a platform for realizing several existing strategies for attacking central conjectures. Haugseng himself should be ideally placed to grasp these opportunities.

In addition to developing new theory, Haugseng has also given important contributions by consolidating different competing theories.

In conclusion, Haugseng is an outstanding young mathematician who combines his profound insight with brute mathematical force within an active and highly competitive field.